Transmission network



March 15, 1932. w. BENNETT 1,849,656

v TRANSMISSION NETWORK Filed June 29, 1929 Ha. I 56.2 L R2 \m,---

196.3 FIG; 4 a n6. 5 C3 1 c; I Z E- Fla. 6

.FREGUENCV RECEIVED QURRENT a 9 W RBENNETT I Arm/W5) Patented Mar. 15,1932 UNITED STATES PATENT OFFICE 1 WILLIAM R. BENNETT, OF JERSEY CITY,NEW JERSEY, ASSIGNOR TO BELL TELEPHONE LABORATORIES, INCORPORATED, OFNEW YORK, N. Y., A CORPORATION OF NEW YORK rnansivrrssron NETWORKApplication filed. June 29,

This invention relates to wave transmission networks, and moreparticularly to artificial lines having broad band frequency selectiveproperties.

In wave transmission systems involving a uniformline, or an artificialline of iterative character, a necessary condition for uniformefiiciency of transmission is that the terminal apparatus with which theline is associated should have an impedance substantially equal to thecharacteristic impedance of the line at all frequencies in the range itis desired to transmit. In certain cases this can be done with greatexactness, but where the line is an iterative structure of reactiveimpedance elements, as in case of a coil loaded line or of a broad bandwave filter,'the condition can be fulfilled only approximately. This isfor the reason'that the characteristic impedances of these structuresare not constant with frequency, except at frequencies remote from thetransmission band limits, and are strongly variable at frequencies closeto the band limits. Whensuch lines are used, as is ordinarily the case,with terminal apparatus of constant resistive impedance, wavereflections occur at the ends of the line and give rise toirregularities in the transmission characteristic. These irregularitiesare most noticeable at points corresponding to the natural frequenciesof the system and may be regarded as due to insufliciency of theclamping of the resonances by the resistive terminal impedance. For manypurposes the resonance effects are of little importance since theirmagnitude is greatly diminished by the effects of dissipation in thebranch impedances, but in other cases, Where the internal dissipation isvery small or where the operating conditions require a high degree ofuniformity, the irregularities due to undamped. resonance may he veryobjectionable.

By this invention broad band selective systems are provided in which thetransmission is free from irregularities of the type described. Thisresult is obtained by departing from the usual iterative type ofstructure and instead, arranging the selective circuit as a taperedlinein which the impedance values of the elements vary from section to1929. Serial No. 374,669.

section in accordance with a prescribed law, as hereinafter described.The structures of the invention comprise artificial lines of the ladder,or series-shunt type which, in schematic form, havegeneral resemblancesto broad band Wave filters of known type, but differ therefrom incertain important respects. One characteristic feature of the structuresis that the series branch impedances increase in value progressivelyfrom both ends towards the center, while the shunt branch impedancesvary in an inverse manner. Viewed in the light of the ordinary wavefilter theory this method of tapering corresponds to varying both thecharacteristic impedance and the Width of the transmission band fromsection to section along the line. The resulting characteristic may beregarded as being due to the internal reflecv tions neutralizing thoseoccurring at the ends of the line, however, the operation of thecircuits of the invention cannot readily be analyzed in terms of theordinary filter theory and the above statement of the principle ofoperation is therefore intended to be merely suggestive.

The nature of the invention and its mode of operation will be more fullyunderstood from the following detailed description and by reference tothe attached drawings, of which Fig. 1 is a schematic diagramillustrating 'the general form of the transmission networks of theinvention; 7

Figs. 2, 3, 4; and 5 show various specific forms of line sectionsapplicable to Fig. 1; Fig. 6 is a diagram illustrating certain points inthe theory of the invention; and

Fig. 7 illustrates the type of transmission characteristic obtained bythe invention.

The circuit illustrated in Fig. 1 comprises terminal resistances R and Rof equal value which may be constituted by resistance elements or mayrepresent the impedances of long transmission lines or of powerconsuming loads, a wave source W included in series with R and anartificial line of the ladder type having n branch impedances Z Z Z Zdisposed alternately in series and in shunt between R and R The branchand the shunt impedances Z Z etc. are given the values,

where R is the common value of resistances R and R X is a frequencyfactor expressing the variation of the impedances with frequency and thefactors a a etc. are numerical coeflicients having the values or ingeneral,

The circuitis shownterminating at the right in a series impedance Z andat the left in a shunt impedance Z n being an even number. It is notnecessary, however, that this particular arrangement be followed; theinvention is not restricted to an even number of branches and each endof the line may terminate in a series or in a shunt impedance regardlessof the termination of the other end. The coefficients a a etc. as givenby Equation 3 are applied to the branches in order, counting from oneend of the line, but it is immaterial which end is used as a startingpoint since the coefiicients are equal in pairs as follows:

(Z =(L 2 n-17 a =a,1 etc. With these proportions the current inresistance R due to an E. M. F. E in series with R is given by theequation E m W where [I] denotes the absolute magnitude, or

modulus value, of the current, and is the modulus value of the frequencyfactor X.

The factor X, which expresses the frequency variation of the branchreactances is an imaginary quantity expressible as an odd rationalfunction of frequency, but its modulus value is a positive realquantity. Numerically X is equal to each of the ratios Z1 5 a .n (1 R (1R a 22 a Z that is, it defines, except for the numerical. coeificients aa etc. the ratio of the values of the series impedances to the terminalresistance, or the ratio of the terminal resistance to the shuntreactances. The general form of the impedance of any reactancestructure, and hence the general form of the frequency factor X, isdiscussed by R. M. Foster, A reactance theorem, Bell System TechnicalJournal, Vol. III, No. 2, April 1924, page 259.

From Equation (4;) it follows that, so long as [XI is less than unity,the value of the current is closely equal to former being defined bynumerical values of X between 1 and -1 From the known properties ofreactances it follows that as IXI increases from zero to unity, thequantity on the right hand side of Equation continuously diminishes fromthe maximum value the rate of diminution being very small at first andthen increasing, but showing no intermediate maxima or minima. Thefrequencies for which X is zero correspond to the resonances of theseries branches and to the anti-resonances of the shunt branches. Thetransmission bands are therefore centered about the resonancefrequencies of the series branches, except when these occur at zerofrequency and infinite frequency.

The determination of the numerical co efficients a a etc, which definethe taper of the line, is a. lengthy and involved mathematicalprocedure, only the major steps of which will be described here.

The ratio of the voltage E of the source to the value to the current Iin'the resistance R may be expanded as :a power series,

in Which the coefficients A A etc. are functions of the coefficients a aetc. of the impedance branches. The simple character of the power seriesis due to the fact that the series impedances and the shunt admittancesare all proportionalto the same frequency function X, thus permittingthe ratio to be expanded as a function ofa single variable.

Equation may be Written in the form ';=A. X.) (X-- X.) Xothe identity isestablished that which is satisfied for A equal to unityand X X etc.respectively equal to the n, nth roots of i1, the sign being taken if nis odd and the sign if n is even. The values of the roots X X etc. maybe Written as follows:

2) 1r +j-sin 2) i when n is odd, the sign being chosen so that the realpart corresponding to the damping constant, is negative.

These roots define the character of the transient, or free, oscillationsof the circuit. The imaginary parts of the roots are proportional to thenatural frequencies and the real parts to the corresponding dampingconstants. If plotted in the complex plane with the real parts asabscisseeand the imaginary parts as ordinates, it will be seen that theroots are represented by equal vectors spaced at equal angles, the anglebetween successive vectors being equal to a dicatedabove possesses theunique transmisr sion characteristics expressed by Equation 4. It is notfreefrom natural or transient oscillations, but in its response toforced oscillations, resonance effects at the several natural periodsare not in evidence. The response to forced oscillations resembles thatof a singly resonant system but is much broader and flatter. In the morecomplex systems of the invention a plurality of transmission bands mayexist, the response characteristics in each band being of the characterof a single broad resonance, free from irregularities at the naturalperiods. The roots in such cases each define a plurality of naturalperiods, one for each transmission range. V

The roots 'X X having been found, the coefficients A A etc. in Equation5 can be determined by standard mathematical processes, thesecoefficients being related to the I roots in the following manner;

2 0 4.2 A0 ie-is the sum of the products taken three A, at a time, andso on.

is the sum of the roots;

is the sum of the products of taken tWo at a time;

' Having determined the necessary values of the coefficients of Equation(5) to make it correspond to the stipulated form of the voltage ratio,the next step is to determine the values of the impedance fcoefiicientsa (1 etc. so that the physical network will have the prescribedcharacteristic. To do this Equation (5) maybe rewritten with the coefficients A A etc. fully expanded, that is,

expressed in terms of the impedance coeffithe one involving thenumerical values of the coefficients A A necessary to obtain 5;:

the prescribed characteristic, and the other involving the coefficientsexpanded as functions of a a etc. Equating the coefficients ofcorresponding powersof X, a sufficient the roots number of relationshipsis .set up to enable {also the individual values of the impedancecoefficients to be found.

By carrying out the steps indicated above for a number of cases,corresponding to progressively increasing values of n from 2 to 10, thegeneral values of the coefficients a (1 etc. given in Equations 3 werearrived at. Further computations in which these values were applied tocircuits having greater numbers of branches have shown that they applyfor all values of n as large as may be desired.

The design of circuits to transmit oscillations between prescribedlimits of frequency may be carried out in a direct manner by giving theseries branches an appropriate structure and assigning such values tothe elements that the reactances have the magnitudes -aR at thespecified limits, the coefiicients a being those appropriate to theorder of the branch. The design of the shunt branches follows from thereciprocal relationship involved. However the following procedure, whichis based on the design of uniform wave filters is generally simpler.

A uniform line of recurrent structure, having series branches ofimpedance 2XR and shunt branches of impedance has its transmission bandlimits defined by the condition which is the same as the conditiondetermining the band limits in the circuits of the invention. Such aline, like the circuits of the invention, is characterized by an inverserelationship between its series and its shunt impedances, the product ofany pair of these impedances being constant and equal to R The designof. wave filters of this type, to which the name constant K has beengiven is described in U. S. Patent 1,227,113, issued May 22, 1917 to Gr.A. Campbell, and in U. S. Patent 1,509,18 issued September 23, 1924: toO. J. Zobel. patents may be used to compute the impedance elements of atypical section of the constant K line having the prescribed cutofffrequencies, and, from the impedances of this typical section, thedesign of the corresponding circuit of the invention may be computed byvirtue of the relationship mentioned above. The impedances of the seriesbranches are equal to half the series impedance of the related constantK section multiplied by the appropriate impedance coefficients ofEquation 3, and the impedances of the shunt branches are found by takingtwice the shunt impedance of the iterative structure and applying theappropriate c0etlicients.

The application of the foregoing principles will be illustrated inconnection with The design formulae of these the values jwL a,XR and 36002 G/T ZX respectively where 0) denotes 211' times the frequency. Ifthe cutoff frequency for which {Xi is unity, be denoted by f it followsthat The form of the transmission characteristic of the low-pass type ofcircuit is shown in Fig. 7 in which the magnitude of the receivedcurrent is plotted against the frequency function f/f The'uniform curves10, 11 and 12, representing the characteristics of circuits having 3, 5and 7 branches respectively, illustrate the high degree of uniformitythat exists throughout the transmission range. For comparison purposesdotted curve 15 shows the value of the voltage ratio in the prototypeconstant K wave filter having five branches. These curves are computedon the assumption that there is no energy dissipation in the energybranches, the effect of dissipation being to add to the curves for thecircuits of the invention a gradual downward slope with frequency, andto reduce the high peak in curve 15.

Fig. 3 shows a typical section of a highpass structure, the seriesimpedance in this case being capacities and the shunt impedancesinductances. Capacity C representing in general the r branch impedancehas the value and inductance L of the (r-l) branch has the value R L31'1 f0 (l5) The frequency factor in this case has the I value X=' 16 Thestructure of Fig. 4 is of the band-pass ill) in Fig. 5.

frequencies, including infinity..

where f if 2 o The design in this case is most readily developed bycamparison with the prototype constant K wave filter, the values of theimpedance elements of which are given by for the shunt branches. Thesevalues lead at once to the impedance values for the circuit of theinvention I l 1 1- 4o" Y 21r(f f 2 a 'i i r f1f2R a, (18) Urfi) I L a12'llf f2 and the series. elements L C as before, being assumed torepresent the W branch and the shunt elements 1/ C, constituting the(1'1) branch of the artificial line.

The circuits of the invention may also be adapted for multi-bandtransmission, an example of a structure of this type being shown Only asingle section is illustrated, but this suflices to show the characterof the branch impedances. The particular structure shown has three passbands, one of which starts at zero frequency. The series impedance Zcomprises an inductance and two anti-resonant circuits all in series,This combination is resonant at three frequencies, including zero, andis anti-resonant at three The shunt impedance Z is inversely related toZ,,, being anti-resonant at the resonance frequencies of Z, and viceversa. It comprises a condenser and two seriesres onant circuits allconnected in parallel. The three transmission bands are located aboutthe three resonance frequencies of the series branch impedance, and maybe placed roughly in their desired positions in the frequency scale byproperly placing the resonance frequencies.

L 4 O4 R 7 Additional bands may be obtained by adding anti-resonantcircuits to the series branch and corresponding resonant circuits to theshunt branch. The formation of the pass bands is illustrated by Fig. 6,which shows the form of the frequency factor X for the circuit of Fig.5. The value of X increases from zero at the first resonant frequency, f=O, to infinity at the first anti-resonant frequency f and passesthrough Zero and infinity at the successive frequencies f f f and f =oo.The transmission ranges are defined by values of X between 1 and 1 andare indicated in the figure by the thickened portions of the horizontalaxis.

Explicit design formulae for multi-band circuits become extremelycomplicated and of little use for practical purposes. It is generallysimpler to proceed towards the final design by a trial method involvingsuccessive approximations, the first step of the process being thedetermination of the constant K prototype, as in the example discussedabove. The form of the branch impedances having been determined for theprescribed number of bands, at trial design of the series impedance maybe made, with the resonant frequencies placed according to the desiredband locations. A plot of the impedance will indicate the positions ofthe band limits and, if these are incorrect, the constants may be variedin successive trials until a good approximation to the desired positionsis obtained. The design of the shunt impedance follows from the inverserelationship, and the final design is obtained by applying the impedancecoefficients a a as already indicated.

What is claimed is:

I 1. A frequency selective transmission linecomprising a plurality ofreactive impedances disposed alternately in series and in shuntrelation, said series impedances being of similar type whereby theirreactances are constantly related at all frequencies, said shuntimpedances being of inverse type to said series impedances, and thereactances of said series impedances and the susceptances of said shuntimpedances being proportional to the quantity 2n where 1" denotestheorder of the impedance sin 1r counting from one end of the line and nis the total number of impedances.

2. A broad band frequency selective system comprising an artificial lineof ladder structure, each series branch of said line including aninductance and each shunt branch includ ing a capacity, the values ofsaid inductances and said capacities being proportional to the quantitysin where r denotes the order of a branch counting from one end of theline, and n is the total number of branches.

4. A broad band frequency selective system comprising a ladder typeartificial line each series branch of which comprises an inductance anda capacity in series and each shunt branch of which comprises aninductance and a capacity in parallel, said series branches being allresonant at a given frequency and said shunt branches being antiresonantat the same frequency, and the inductances of said series branches andthe capacities of said series branches being proportional to thequantity where 1" denotes the order of a branch counting from one end ofthe line, and a is the total number of branches.

5. In combination with a wave source and a terminal load having equalresistive impedances, a network in accordance with claim 1 the branchimpedances of which are proportioned with respect to the resistance ofthe terminal impedances and to the limiting frequencies of a preassignedrange whereby transmission through the network in the preassignedfrequency range is substantially free from irregularities due to wavereflection at the terminals.

6. A wave transmission network comprising a plurality of reactiveimpedances disposed alternately in series and in shunt to constitute amulti-section ladder type line, said series impedances being of similartype whereby their reactances are constantly related at all frequencies,said shunt impedances being of inverse type to said series impedances,and the reactances of said series impedances and the susceptances ofsaid shunt impedances increasing symmetrically from both ends of theline towards the center.

7. A wave transmission network comprising a plurality of reactiveimpedances disposed alternately in series and in shunt to constitute amulti-section ladder type line, said series impedances being of similartype, said shunt impedances being of inverse type to said seriesimpedances, and the reactances of said series impedances and thesusceptances of said shunt impedances increasing from both ends of theline towards the center, the rate of increase diminishing towards thecenter.

8. A broad-band wave filter comprising a plurality of sections, thetransmission bands of said sections having a frequency range in commonand said sections being so proportioned that their respective bandwidths vary progressively from both ends towards the center of thenetwork.

9. A broad-band wave filter comprising a plurality of sections, thetransmission bands of said sections having a frequency range in commonand said sections being so proportioned that their respective bandwidths diminish progressively from both ends towards the center of thenetwork.

10. A transmission network comprising a plurality of sections, each ofsaid sections being adapted to transmit selectively oscillations in abroad frequency range and said sections being so proportioned withrespect to each other that the transmission bands of adjacent sectionshave a frequency range in common and vary progressively in width fromboth ends towards the center of the network.

11. The method of eliminating transmission irregularities in abroad-band multisection wave filter which comprises progressivelyincreasing the transmission band widths of adjacent sections from thecenter of the filter towards both ends.

12. A low'pass wave filter comprising aplurality of sections, eachsection including series inductance and shunt capacity, the in ductancesand capacities of said sections being so proportioned that the cut-offfrequencies of the respective sections increase progressively from thecenter of the filter towards both ends.

13. The method of eliminating transmission irregularities in abroad-band multi-section wave filter which comprises progressivelyincreasing the transmission band widths of adjacent sections from thecenter of the filter towards both ends and varying the values of thenominal characteristic impedances from one section to another.

14. A broad-band wave filter comprising a plurality of sections, thetransmission bands

